Find the unit vector in the direction of vector PQ, where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively.
The unit vector in the direction of vector PQ is (1/)(î + ĵ + k̂) or (1/)î + (1/)ĵ + (1/)k̂.
Explanation
This question tests the student's understanding of finding a vector joining two points and then converting it to a unit vector. The solution follows the method shown in Example 10 of the textbook, where the vector joining two points P and Q is found using the formula PQ = (x₂ - x₁)î + (y₂ - y₁)ĵ + (z₂ - z₁)k̂. The unit vector is then obtained by dividing the vector by its magnitude.
Solution Steps
Step 1: Find the vector PQ. Given P(1, 2, 3) and Q(4, 5, 6), the vector PQ is: PQ = (4 - 1)î + (5 - 2)ĵ + (6 - 3)k̂ = 3î + 3ĵ + 3k̂
Step 2: Find the magnitude of PQ. |PQ| = = = = 3
Step 3: Find the unit vector. Unit vector in the direction of PQ = PQ/|PQ| = (3î + 3ĵ + 3k̂)/(3) = (1/)(î + ĵ + k̂)